## Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system

JOSA A, Vol. 20, Issue 12, pp. 2281-2292 (2003)

http://dx.doi.org/10.1364/JOSAA.20.002281

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### Abstract

Taking the classical Ignatowsky/Richards and Wolf formulas as our starting point, we present expressions for the electric field components in the focal region in the case of a high-numerical-aperture optical system. The transmission function, the aberrations, and the spatially varying state of polarization of the wave exiting the optical system are represented in terms of a Zernike polynomial expansion over the exit pupil of the system; a set of generally complex coefficients is needed for a full description of the field in the exit pupil. The field components in the focal region are obtained by means of the evaluation of a set of basic integrals that all allow an analytic treatment; the expressions for the field components show an explicit dependence on the complex coefficients that characterize the optical system. The electric energy density and the power flow in the aberrated three-dimensional distribution in the focal region are obtained with the expressions for the electric and magnetic field components. Some examples of aberrated focal distributions are presented, and some basic characteristics are discussed.

© 2003 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(050.1960) Diffraction and gratings : Diffraction theory

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(110.2990) Imaging systems : Image formation theory

**Citation**

Joseph J. M. Braat, Peter Dirksen, Augustus J. E. M. Janssen, and Arthur S. van de Nes, "Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system," J. Opt. Soc. Am. A **20**, 2281-2292 (2003)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-12-2281

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### References

- B. H. W. Hendriks, J. J. H. B. Schleipen, S. Stallinga, and H. van Houten, “Optical pickup for blue optical recording at NA=0.85,” Opt. Rev. 8, 211–213 (2001).
- H. P. Urbach and D. A. Bernard, “Modeling latent-image formation in the photolithography, using the Helmholtz equation,” J. Opt. Soc. Am. A 6, 1343–1356 (1989).
- V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Tr. Opt. Inst. 1 (4), 1–36 (1919).
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
- A. J. E. M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
- J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
- P. Dirksen, J. J. M. Braat, A. J. E. M. Janssen, and C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr. Microfabr. Microsyst. 2, 61–68 (2003).
- J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Retrieval of aberrations from intensity measurements in the focal region using an extended Nijboer–Zernike approach,” manuscript available from the author, j.j.m.braat @tnw.tudelft.nl.
- N. R. Heckenberg, T. A. Nieminen, M. E. J. Friese, and H. Rubinsztein-Dunlop, “Trapping microscopic particles with singular beams,” in International Conference on Singular Optics, M. S. Soskin, ed., Proc. SPIE 3487, 46–53 (1998).
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
- L. Allen, J. Courtial, and M. J. Padgett, “Matrix formulation for the propagation of light beams with orbital and spin angular momenta,” Phys. Rev. E 60, 7497–7503 (1999).
- W. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986).
- S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001).
- M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
- B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1942).
- P. Török, P. Varga, and G. Nemeth, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
- A. J. E. M. Janssen, “Stable representation of Zernike polynomials,” Internal Rep. NL TN, 2001/263 (Philips Research Laboratories, Eindhoven, The Netherlands, 2002).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
- G. Szegő, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).
- F. G. Tricomi, Vorlesungen über Orthogonalreihen (Springer, Berlin, 1955).
- G. E. Andrews, R. Askey, and R. Roy, Special Functions (Cambridge U. Press, Cambridge, UK, 1999).

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